I have to correct part of my recent reply to a posting of John Huesman:

Another option, which is available also for non-solvable groups G,
is to make use of the isomorphism between H^1(G,L) and H^0(G,R^n/L).
This isomorphism holds whenever G is finite, for the same reasons
as the isomorphism between H^2(G,L) and H^1(G,R^n/L).

This isomorphism relies on H^0(G,R^n) being zero. I was under the
impression that this was always the case, but Derek Holt pointed
out to me that this is not true. In fact, H^0(G,R^n) is zero if and
only if there is no vector in R^n left invariant by G. Clearly,
there are groups G with invariant vectors, and for those the method
I proposed does not work. I am sorry for this confusion.

As a little compensation I am offering routines, with the help of which
one can complete the solution proposed by Derek Holt. He posted
functions to compute matrices whose integral kernel, resp. row space,
generate the cocycles and coboundaries of the cohomology group
H^1(G,Z^n). The functions appended below can be used to compute a
basis of the integral row space and integral kernel of an integer
matrix. There is also a function to compute the structure
of the quotient of two row modules over the integers. With these
functions, and the ones posted by Derek Holt, H^1(G,Z^n) is then
computed as follows: